Real Analysis Answers

If a sequence is bounded, then it has at least two convergent subsequences. Check whether the given statement is true of false. Give a counterexample in support of your answer.

Give an example for each of the following. (10)

i) A set in 

R

with a unique limit point.

ii) A set in 

R

whose all points except the one are its limit points.

iii) A set having no limit point.

iv) A set 

S

with 

S°=S.



v) A bijection from 

N odd

to 

Z

Which of the following statement s are true and which are false?Justify your answer with a short proof or a counter example.

i) If x and y are real numbers such that x<y,then x^2<y^2.

ii)for every finite set S,sup S€S.

iii) There exists an interval with infimum and supremum equal.

iv)The sequence (1,1/2,1/3,1/4,....) is unbounded.

v) If a sequence is bounded ,then it has at least two convergent subsequences.

If a_(1)<**<a_(n) find the minimum value of f(x)=sum_(i=1)^(n)(x-a_(i))^(2) .

State and prove weierstrass M- test

Let D is a subset of R² be such that (a,infinity)×(c,infinity) subset of D for some a,c element of R and.let f:D to R be a function .prove that lim_{(x,y)tends to (infinity, infinity)}f(x,y) exists.if and only if there is a l element of R satisfying (epsilon-(alpha,beeta) condition.

Prove the following result:

A function f that is decreesing on [a,b] is integrable on [a,b].

Let {f_n} be a sequence of functions defined on S .show.that there exists a.function f such that f_n converges to f uniformly on S if and.only if the caught condition is satisfied

Define bounded and unbounded variation. Show that every function which is of bounded variation is bounded

Define uniform convergence of sequence of functions. Give an example